Solution ch 6 PDF

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Summary

Chapter 6. Centre of Gravity, Centroid and Moment of Inertia Center of Gravity, Center of Mass, and Centroid for a Body 6-1. Locate the center of mass of the homogeneous rod bent in the form of a parabola. Prob. 9-1 Prob. 6-2 Solution x   xdL ,  dL y  L L  ydL ,  dL z  L L  zdL  dL L L dL ...

Description

Chapter 6. Centre of Gravity, Centroid and Moment of Inertia Center of Gravity, Center of Mass, and Centroid for a Body 6-1. Locate the center of mass of the homogeneous rod bent in the form of a parabola.

Prob. 9-1

Prob. 6-2

Solution x 

x ,  dL

y 

L

L

 dL

z 

,

L

 dL L

dL  dy  dx  y   1dx (x, y)C  (0, 0.912) m 2

2

2

6-2. Determine the weight of the rod and the distance y to its center of mass. The rod has a mass per unit length of 0.5 kg/m. Solution dL  dy 2  dx 2  y 2  1dx

L

1



0

y 2  1dx  1.44 m

m  0.5L  0.72 kg (x, y)C  (0.5457, 0.4465)m

Ch 06

1

6-3. Locate the center of gravity of the thin homogeneous cylindrical shell. xG  0 m yG  h / 2 zG  2 2a zG   0.90a 

Prob. 6-3 dA  h  dL  h  a  d

A ha  /4

zG

 

zG

 

 /4

/4



 /4

d   ha /2

zdA

A z  a cos   /4

 /4

zG 

a cos had

2 2a



A  0.90a

6-4. Determine the location (x , y ) of the centroid of the triangular area. dA  y  dx  mx  dx A



a 0

dA 

1 ma 2 2

 xdA  x (mxdx ) 2   a 3 A A  ydA  (mx )(mx )dx 2 yG    ma A A 3

xG 

Prob . 6- 4

Ch 06

2

Composite Bodies 6-5. Locate the centroid (x , y ) of the shaded area. dA  (y1  y2 )dx A

4

 (y 0

1

 y 2)dx

x  y 2)

y

x A y yG  A

xG 

dA  A (y 1  y 2 )dA A

(xG, yG)  (1.8, 1.8) m

Prob. 6-5

6-6. The gravity wall is made of concrete. Determine the location (x , y ) of the center of gravity G for the wall.

 xi Ai  2.22  Ai  yiAi yG   1.41  Ai xG 

Prob. 6-6 A1  0.4 * (1.2  2.4), (x1, y1 )  (1.8, 0.2) A2  12 * 1.8 * 3,

(x2, y2 )  (0.6  2 * 1.8 / 3, 0.4  3/ 3)

A3  3 * 0.6,

( x3, y3 )  (0.6  1.8  0.3,0.4  1/ 2 * 3)

A4  12 * 3 * 0.6,

( x4, y4 )  (0.6  2.4  1/ 3 * 0.6,0.4  2 / 3 * 3)

Ch 06

3

Theorems of Pappus and Guldinus 6-7. Determine the surface area and the volume of the ring formed by rotating the square about the vertical axis. surface area S  2 dC L dC  b , L  4a

S  2 b(4a)  8 ab volume of the ring V  2 dC A dC  b, A  a 2 V  2ba2

Prob. 6-7 6-8. A circular belt has an inner radius of 600 mm and the cross-sectional area shown. Determine the volume of material required to make the belt. 1

2

A1  12  25  75  A3 ,

3

( x 1, y1)  (x 1,

2 3

 75)

( x 3 , y3 )  ( x 3 ,

2 3

 75)

A2  50  75 ( x 2, y2)  (0,

1 2

 75)

 xi Ai 0  Ai  y iAi yG   41.67 mm  Ai xG 

Prob. 6-8 A   Ai V  2 (600  yC )A  22.68  1003 mm3 V  2 (600  yC )A  22.68 dm3

6-9. Determine the surface area of the tank, which consists of a cylinder and hemispherical cap. 6-10. Determine the volume of the tank, which consists of a cylinder and hemispherical cap.

Ch 06

4

Prob. 6-9/10

Moments of Inertia for Areas 6-11. Determine the moment of inertia of the triangular area about the x axis. 6-12. Determine the moment of inertia of the triangular area about the y axis.

dA

Prob. 6-11/12

Prob. 6- 13

6-13. Determine the radius of gyration of the shaded area about the y axis. 6-14. Determine the moment of inertia of the Z-section about the x and y axis.

Ch 06

5

dA dA

Prob. 6-15

Prob. 6- 14 6-15. Determine the product of inertia of the shaded area with respect to the x and y axes. Product of Inertia for an Area 6-16. Determine the product of inertia of the quarter circular area with respect to the x and y axes. Then use the parallel-axis theorem to determine the product of inertia about the centroidal x' and y' axes.

Prob. 6- 16

Ch 06

6...